A frustum of a right circular cone has a diameter of base and top 20 cm and 12 cm, respectively and a height of 10 cm. Find the area of its whole surface and volume.

Answer

To find the area of the whole surface and the volume of the frustum of a right circular cone, we can follow these steps:

### Step 1: Identify the dimensions of the frustum
- Diameter of the base (d1) = 20 cm
- Diameter of the top (d2) = 12 cm
- Height (h) = 10 cm

### Step 2: Calculate the radii of the base and the top
- Radius of the base (R1) = d1 / 2 = 20 cm / 2 = 10 cm
- Radius of the top (R2) = d2 / 2 = 12 cm / 2 = 6 cm

### Step 3: Calculate the volume of the frustum
The formula for the volume (V) of a frustum of a cone is:
\[ V = \frac{1}{3} \pi h (R1^2 + R2^2 + R1 \cdot R2) \]
Substituting the values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 10 \times (10^2 + 6^2 + 10 \times 6) \]
Calculating the squares and products:
\[ = \frac{1}{3} \times \frac{22}{7} \times 10 \times (100 + 36 + 60) \]
\[ = \frac{1}{3} \times \frac{22}{7} \times 10 \times 196 \]
Calculating further:
\[ = \frac{220 \times 196}{21} \]
Calculating \(220 \times 196\):
\[ = 43120 \]
So,
\[ V = \frac{43120}{21} \approx 2053.33 \, \text{cm}^3 \]

### Step 4: Calculate the slant height (l) of the frustum
The formula for the slant height (l) is:
\[ l = \sqrt{(R1 - R2)^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{(10 - 6)^2 + 10^2} \]
Calculating:
\[ = \sqrt{4^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116} = 2\sqrt{29} \, \text{cm} \]

### Step 5: Calculate the total surface area (A) of the frustum
The formula for the total surface area is:
\[ A = \pi (R1 + R2) l + \pi R1^2 + \pi R2^2 \]
Substituting the values:
\[ A = \frac{22}{7} \times (10 + 6) \times (2\sqrt{29}) + \frac{22}{7} \times 10^2 + \frac{22}{7} \times 6^2 \]
Calculating:
\[ = \frac{22}{7} \times 16 \times (2\sqrt{29}) + \frac{22}{7} \times 100 + \frac{22}{7} \times 36 \]
\[ = \frac{22}{7} \times (32\sqrt{29} + 136) \]

### Final Results
- Volume of the frustum: \( \approx 2053.33 \, \text{cm}^3 \)
- Total surface area: \( \frac{22}{7} (32\sqrt{29} + 136) \, \text{cm}^2 \)